https://file.aiquickdraw.com/mathbot/file/1728951774124yzfh74i9.jpg

The problem asks to **factor and simplify** the following trigonometric expression:

\[
10 \cos^2{x} + \cos{x} - 11
\]

This looks like a quadratic equation in terms of \( \cos{x} \). Let's solve it step by step.

---

### Step 1: Substitute \( u = \cos{x} \)
This makes the equation:

\[
10u^2 + u - 11 = 0
\]

### Step 2: Solve the quadratic equation
Use the quadratic formula:

\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 10 \), \( b = 1 \), and \( c = -11 \). Substituting these values:

\[
u = \frac{-1 \pm \sqrt{1^2 - 4(10)(-11)}}{2(10)}
\]

\[
u = \frac{-1 \pm \sqrt{1 + 440}}{20} = \frac{-1 \pm \sqrt{441}}{20} = \frac{-1 \pm 21}{20}
\]

This gives two solutions:

1. \( u = \frac{-1 + 21}{20} = \frac{20}{20} = 1 \)
2. \( u = \frac{-1 - 21}{20} = \frac{-22}{20} = -\frac{11}{10} \)

---

### Step 3: Back-substitute \( u = \cos{x} \)
- For \( \cos{x} = 1 \), the solution is:

  \[ x = 0 + 2k\pi \quad \text{(where \( k \) is any integer)} \]

- For \( \cos{x} = -\frac{11}{10} \), this is **impossible** because \( \cos{x} \) is bounded by \( -1 \leq \cos{x} \leq 1 \).

---

### Step 4: Factor the original expression
Using the valid solution \( \cos{x} = 1 \), the factorization of the quadratic equation corresponds to:

\[
10(\cos{x} - 1)(\cos{x} + \frac{11}{10}) = 0
\]

However, since \( \cos{x} = -\frac{11}{10} \) is not a valid solution, the only valid factorization relevant to real solutions is:

\[
10(\cos{x} - 1) = 0
\]

---

### Final Answer:
The solution to the equation is:

\[
\boxed{\cos{x} = 1}
\]

Thus, \( x = 0 + 2k\pi \), where \( k \) is any integer.

---

Do you need further clarification or have any questions? 

---

Here are **5 related questions** for deeper understanding:
1. How can we solve other trigonometric equations with quadratic forms?
2. What are the bounds of \( \cos{x} \) and \( \sin{x} \), and why do they matter?
3. What is the difference between factoring and solving a trigonometric equation?
4. How does the quadratic formula work in more complex trigonometric contexts?
5. Can other substitution methods help with trigonometric equations?

**Tip:** Always check the domain constraints when dealing with trigonometric functions to avoid extraneous solutions!

Factor and simplify: 10 cos^2(x) + cos(x) - 11

Learn how to factor and simplify the trigonometric expression 10 cos^2(x) + cos(x) - 11 using quadratic methods. This problem covers key trigonometric and algebraic concepts, including factoring and the quadratic equation formula. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation